A084231 Numbers k such that the root-mean-square value of 1, 2, ..., k, i.e., sqrt((1/k)*Sum_{j=1..k} j^2), is an integer.

1, 337, 65521, 12710881, 2465845537, 478361323441, 92799630902161, 18002650033695937, 3492421306906109761, 677511730889751597841, 131433783371304903871537, 25497476462302261599480481, 4946378999903267445395341921, 959572028504771582145096852337

Offset

1,2

Comments

Equivalently, numbers k such that sqrt((k+1)*(2*k+1)/6) is an integer.

Links

Indranil Ghosh, Table of n, a(n) for n = 1..437

Peter Khoury and Gerard D. Koffi, Continued fractions and their application to solving Pell’s equations (2009)

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (195,-195,1).

Formula

a(n) = ((7/2 + 2*sqrt(3))*(97 + 56*sqrt(3))^n + (7/2 - 2*sqrt(3))*(97 - 56*sqrt(3))^n - 3)/4.

a(n) = (floor((7/2 + 2*sqrt(3))*(97 + 56*sqrt(3))^n) - 2)/4.

a(n+3) = 195*(a(n+2) - a(n+1)) + a(n).

G.f.: x*(1+142*x+x^2)/((1-x)*(1-194*x+x^2)).

a(n) = ((7 - 4*sqrt(3))^(1+2n) + (7 + 4*sqrt(3))^(1+2n) - 6)/8. - Peter Pein (peter.pein(AT)dordos.de), Mar 03 2005

a(n) = 195*a(n-1) - 195*a(n-2) + a(n-3), with a(0)=0, a(1)=1, a(2)=337, a(3)=65521. - Harvey P. Dale, Jul 14 2011

Example

337 is in the sequence because sqrt((1/337)*Sum_{k=1..337} k^2) is an integer (195=A084232(1)).

Mathematica

a[n_]:=Expand[((7-4 Sqrt[3])^(1+2n)+(7+4 Sqrt[3])^(1+2n)-6)/8] (* Peter Pein *)
CoefficientList[Series[x (1+142x+x^2)/((1-x)(1-194x+x^2)), {x, 0, 30}], x] (* or *) Join[{0}, LinearRecurrence[{195, -195, 1}, {1, 337, 65521}, 30]] (* Harvey P. Dale, Jul 14 2011 *)

Crossrefs

Cf. A084232.

Sequence in context: A263865 A184202 A194478 * A243483 A234625 A226539

Adjacent sequences: A084228 A084229 A084230 * A084232 A084233 A084234

Keyword

nonn,easy

Author

Ignacio Larrosa Cañestro, May 20 2003

Extensions

One more term from Peter Pein (peter.pein(AT)dordos.de), Mar 03 2005

Status

approved